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**HallsofIvy** this doesn't quite make sense: 'first vector is "r" and the second "theta"' because those are numbers, not vectors. I think what you mean rather is that we have **different** basis vectors at each point. One of the basis vectors, $\displaystyle \vec{v_r}$, at (x, y), is the unit vector whose direction is **away** from the origin, along the line y= x, and the other, $\displaystyle \vec{v_\theta}$ is the unit vector perpendicular to that in the direction of increasing angle. Those. I presume, are your "er" and "e(theta)", respectively.

But I don't know what you mean by "Er" and "i"? did you mean "er" again? And is "i" the unit vector in the x-direction?